From OeisWiki
(Redirected from
Unit)
There are no approved revisions of this page, so it may
not have been
reviewed.
This article page is a stub, please help by expanding it.
An element of a
set is a
unit if its
multiplicative inverse belongs to that set. (The set of units form a multiplicative group.)
Algebraic integer units have
complex norm and are thus complex
roots of unity.
Units in quadratic integer rings
Imaginary
quadratic integer rings have only a few units (see
Theorem I1). The
primitive root of unity of degree

is used in the following table.
Units of imaginary quadratic integer rings
 Units

≤ @opsp@−@opsp@5
 {( − 1)^{0},( − 1)^{1}} = {1, − 1}

@opsp@−@opsp@3


@opsp@−@opsp@2
 {( − 1)^{0},( − 1)^{1}} = {1, − 1}

@opsp@−@opsp@1
 {i^{0},i^{1},i^{2},i^{3}} = {1,i, − 1, − i}

Real quadratic integer rings have infinitely many units (see
Theorem R1), which are all powers of each other, some multiplied by
. For that reason, the table below can't be complete like the table above.
An inefficient way to find a unit of a real quadratic integer ring
other than
or
is to try each positive value of
starting with
and going up until
is an integer.
Units of real quadratic integer rings
 Units

2


3


4
 N/A

5
 (golden ratio)

6


7


8
 N/A, but note that 3^{2} − 8(1^{2}) = 1

9
 N/A

10


11


12
 N/A, but 7^{2} − 12(2^{2}) = 1

13


14


15


16
 N/A

17


18
 N/A, but 17^{2} − 18(4^{2}) = 1

19


20
 N/A, but 9^{2} − 20(2^{2}) = 1

21


22


23


24
 N/A, but 5^{2} − 24(1^{2}) = 1

25
 N/A

26


27
 N/A, but 26^{2} − 27(5^{2}) = 1

28
 N/A, but 127^{2} − 28(24^{2}) = 1

29


33


Examples
 The units of are
{i 0, i 1, i 2, i 3} = {1, i, −1, − i} 
, where . (See Gaussian integers.)
 The units of are .
 The units of are .
 The units of are .
See also